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Josh is looking to buy a second hand car. Josh is risk averse. The cars all sell for $3,500 but there is an 6 percent chance that the car will be worth $1,000, a 53 percent chance it will be worth $3,000 or a 41 percent chance that the car is worth an unknown amount. What must the unknown value be if the expected value is to equal $3,500? Give your answer to the nearest whole dollar (with no decimal points, spaces, $ signs, or commas in your answer).

Question

Josh is looking to buy a second hand car. Josh is risk averse. The cars all sell for 3,500butthereisan6percentchancethatthecarwillbeworth3,500 but there is an 6 percent chance that the car will be worth 1,000, a 53 percent chance it will be worth 3,000ora41percentchancethatthecarisworthanunknownamount.Whatmusttheunknownvaluebeiftheexpectedvalueistoequal3,000 or a 41 percent chance that the car is worth an unknown amount. What must the unknown value be if the expected value is to equal 3,500? Give your answer to the nearest whole dollar (with no decimal points, spaces, $ signs, or commas in your answer).

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Solution

To solve this problem, we first need to understand the concept of expected value. The expected value is the average outcome of a random event given all possible outcomes and their probabilities.

In this case, the expected value is given as 3,500.ThismeansthatifJoshweretobuymanycars,theaveragevalueofthesecarswouldbe3,500. This means that if Josh were to buy many cars, the average value of these cars would be 3,500.

The problem also gives us the probabilities and values of two of the possible outcomes. There is a 6% chance that the car will be worth 1,000anda531,000 and a 53% chance that it will be worth 3,000.

We can calculate the expected value contribution of these two outcomes by multiplying the value of the outcome by its probability.

For the 1,000car,thiswouldbe0.061,000 car, this would be 0.06 * 1,000 = 60.Forthe60. For the 3,000 car, this would be 0.53 * 3,000=3,000 = 1,590.

Adding these two amounts together, we get $1,650.

This means that the expected value contribution of the unknown outcome must be 3,5003,500 - 1,650 = $1,850.

The problem also tells us that the probability of the unknown outcome is 41%.

To find the value of the unknown outcome, we can divide the expected value contribution of the unknown outcome by its probability.

So, 1,850/0.41=1,850 / 0.41 = 4,512.20.

Rounding to the nearest whole dollar, the unknown value must be $4,512.

This problem has been solved

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